Stats: Data and Models
Fifth Edition, Global Edition
Chapter 9
Multiple
Regression
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Slide - 1
Section 9.1 What is Multiple
Regression?
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Slide - 2
When Linear Regression Is Not Enough
• R 2  67.8% for Waist Size and %Body Fat
• 68% of the variation in %Body Fat is accounted
for.
• What about the other 32%?
• Could include other variables such as Height
• A regression with two or more predictor variables
is called a multiple regression.
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1
Multiple Regression
• For a simple regression, with one independent
variable, the least squares line makes residuals as
small as possible.
• For multiple regression, the regression equation
still makes the residuals as small as possible.
• Calculations difficult. Use Software.
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The Results (1 of 2)
• R 2  71.3% is
the percent of
variation in %Body
Fat accounted for by
the multiple
regression equation.
• s = 4.460 is the standard deviation of the residuals.
• df  Sample size  # variables  250  3  247
• Coefficient:
%𝐵𝑜𝑑𝑦𝐹𝑎𝑡
3.10
1.77𝑊𝑎𝑖𝑠𝑡
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0.602 𝐻𝑒𝑖𝑔ℎ𝑡
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The Results (2 of 2)
• SE(Coeff)
represents the
standard error for
each of the three
coefficients.
• The t-ratio and the P-value are the test statistic
and P-value for each coefficient.
• Multiple regression is a versatile tool that can put
together many variables to create a useful model.
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2
What’s Different in Multiple Regression
• Meaning of coefficients has changed in a subtle
way.
• Is an extraordinarily versatile calculation,
underlying many widely used statistics methods.
• Offers a glimpse into statistical models that use
more than two quantitative variables. Models that
use several variables can be a big step toward
realistic and useful modeling of complex
phenomena and relationships.
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Slide - 8
Section 9.2 Interpreting Multiple
Regression Coefficients
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Slide - 10
Height and %Body Fat Only
• Can we use Height to
predict %Body Fat?
• The scatterplot
suggests we can’t.
• Multiple regression for
Height and %Body Fat had very small P-value
• How can we use waist size to help explain %Body
Fat?
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3
Multiple Regression: Height
• Consider only those
who have Waist 36 to
38 inches (blue dots).
• Clear linear trend
• For each small range of Waist, we get a linear
trend.
• For men with a 36-inch waist, an extra inch of
height accounts for 0.60% less body fat.
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Slide - 12
Multiple Regression: Waist
• No longer means what it
did in simple regression.
• Waist: how body fat
percent of men of same
height tends to vary with
waist size.
• Coefficient was 1.7, now is 1.77.
• Hasn’t changed much.
• Meaning has changed!
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Slide - 13
Multiple Regression: Coefficients
• Can’t assume coefficients will stay the same.
• Coefficients change
• Often in unexpected ways
• Even changing signs
• Be alert for a change in value.
• Be alert for a change in meaning.
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4
Multiple Regression Model: Coefficients
𝑦
𝑏
𝑏 𝑥
⋯
𝑏 𝑥
• No simple relationship between y and x j , yet b j
in a multiple regression may be quite different from
zero
• Strong two-variable relationship between y and
x j , yet b j in a multiple regression to be almost zero
• Strong two-variable relationship between y and
x j , yet b j can be opposite in sign in a multiple
regression.
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Slide - 15
Section 9.3 The Multiple
Regression Model – Assumptions
and Conditions
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Slide - 16
The Model
• The model for multiple regression is similar to
simple regression:
• Residuals
𝑦
𝑏
𝑏 𝑥
𝑒
𝑦
𝑦
⋯
𝑏 𝑥
• Easy to extend to more than two predictor variables
– Just add more terms.
• Assumptions and conditions similar to simple
regression
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5
Linearity Assumption
Straight Enough Condition
• We must check the scatterplot for each of the
predictor variables.
• Do not need the scatterplots to show any
discernible slope, but should be reasonably
straight
• Cannot have bends, or other nonlinearity
• Can be easier to look at the plot of residuals
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Plot of the Residuals
• No pattern with respect to predicted values
• Randomly scattered and no patterns or clumps
when plotted against predicted values
• Spread should be uniform when plotted against
any of the x’s or against the predicted values
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Equal Variance Assumption (1 of 2)
• Same variability of the errors for all values of each
predictor
• Does the Plot Thicken? Condition: The spread
around the line must be nearly constant.
• Be alert for “fan” shaped pattern
• or other tendency for variability to grow or shrink
in one part of the scatterplot
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6
Equal Variance Assumption (2 of 2)
• Residual plots below show no pattern. Equal
variance assumption satisfied.
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Check the Residuals
• Errors have a distribution that is:
– Unimodal
– Symmetric
– Without outliers
• Look at histogram or Normal probability plot of residuals
• Assumption is less important as sample size increases
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Slide - 22
Checks of Conditions
1. Straight Enough Condition: scatterplots of y-variable against
each x-variable
2. If straight enough, fit multiple regression model
3. Find the residuals and predicted values.
4. Scatterplot of the residuals against predicted values:
patternless, no bends, no thickening
5. How were data collected? Random? Represent identifiable
population? Time? check independence
6. Histogram of residuals: unimodal, symmetric, without
outliers
7. If conditions check out, interpret regression model, and make
predictions.
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